HandsOn 17 - Modeling a Rough Surface

As part of your data analysis you will measure the fractal dimension of the surface, or perimeter, of the bacterial colony. The question is whether the colony interface may reflect social interactions, or whether the spread of the colony is effectively random.

A random surface can be generated by the following random walk exercise. On a piece of graph paper, draw a horizontal line across the middle. This is your y = 0 line. Make a mark at the leftmost point of the axis. This is your origin (x = 0, y = 0). Move your marker over one unit to x = 1. Flip a coin. If heads results, move up to y = +1; if tails, move down to y = -1. Move your marker over to x = 2. Flip the coin. If it is heads, move y up one; if tails, move y down one. Repeat this process moving your marker over one unit on the x-axis each time (the x-axis represents time), and up or down one unit in the y-direction, depending on the result of the coin flip. For example, if you flip THHTTTHTHHT then the connected points make a graph like Figure .

Figure 5.1: Generation of a random surface using coin flips, heads (H) or tails (T). This surface results from the sequence THHTTTHTHHT.

The result is a random surface. Connect your points with dark magic marker, scan it into the computer with a contrast setting so that the grid vanishes, and find the dimension using the Fractal Dimension program.

 Q5.4: What is the fractal dimension of your rough surface?

 Q5.5: What happens to the surface if we change to the following rule: flip two coins, move up only if you get two heads and move down only if you get two tails.

A common model of molecules in gases and liquids shows particles constantly in motion, dancing around due to thermal disturbances. In a uniform gas or liquid nothing changes, on average, because of this motion; on average, equal numbers of molecules move in and out of a given volume. Since the motion is random, like that of a random walker, there is no "current,'' no net directed flow of molecules in a given direction. This is a picture of a medium that is "at equilibrium.''

Now extend this mental model to the case in which a concentration difference exists. For example, suppose we have a barrier between two containers of water. The container on the left has salt dissolved in the water; the one on the right is pure distilled water. We now raise this barrier carefully, so as not to cause fluid motion.

 Q5.6: What happens to the saltiness on the two sides of the unified chamber? Is there a net motion of salt ions from one side to the other? Why? Does the random walk of the molecules in solution contribute to this?

When a concentration difference occurs between regions of the same medium, that medium is no longer in equilibrium. The result is a net diffusion of particles from the high-concentration region to the low-concentration region.

 Q5.7: As the bacterial colony grows, it eats the nutrients that are located closest to it (near its interface with the gel). This results in a lowered concentration of nutrient near the interface. But the agar gel began as a homogeneous medium with a uniform concentration of nutrients. What happens to the nutrients in the gel? Do they diffuse toward the depleted region? Will the colony still be able to continue to feed even if it does not grow?

In HandsOn 18 you will be growing bacteria under various nutritional conditions. You will prepare agar plates to grow the bacteria and you will create streak plates to produce individual colonies of genetically identical bacteria.

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